Existence and approximating of common best proximity points of relatively nonexpansive mappings via Ishikawa iteration method. (English) Zbl 07884500
Summary: In this article, we study the existence of a common best proximity points for a finite class of cyclic relatively nonexpansive mappings in the setting of Busemann convex spaces. In this way, we extend the main results given in [A. A. Eldred and V. S. Raj, Acta Sci. Math. 75, No. 3–4, 707–721 (2009; Zbl 1212.54116)] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. We then present a strong convergence theorem of a common best proximity point for a pair of cyclic mappings in uniformly convex Banach spaces by using the Ishikawa iterative process.
MSC:
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 90C48 | Programming in abstract spaces |
| 46B20 | Geometry and structure of normed linear spaces |
Keywords:
best proximity point; fixed point; cyclic relatively nonexpansive; uniformly convex Banach space; iterative sequenceCitations:
Zbl 1212.54116References:
| [1] | G.C. Ahuja, T.D. Narang, S. Trehan, Best approximation on convex sets in a metric space, J. Approx. Theory, 12(1974), 94-97. · Zbl 0284.41012 |
| [2] | M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin Heidelberg, 1999. · Zbl 0988.53001 |
| [3] | H. Busemann, The Geometry of Geodesics, Academic Press, New York, 1955. · Zbl 0112.37002 |
| [4] | A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171(2005), 283-293. · Zbl 1078.47013 |
| [5] | A.A. Eldred, V. Sankar Raj, On common best proximity pair theorems, Acta Sci. Math. (Szeged), 75(2009), 707-721. · Zbl 1212.54116 |
| [6] | A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl, 323(2006), 1001-1006. · Zbl 1105.54021 |
| [7] | R. Espinola, P. Lorenzo, Metric fixed point theory on hyperconvex spaces: Recent progress, Arab J. Math., 1(2012) 439-463. · Zbl 1273.54052 |
| [8] | R. Espinola, O. Madiedo, A. Nicolae, Borsuk-Dugundji type extensions theorems with Busemann convex target spaces, Annales Academie Scientiarum Fennica, 43(2018), 225-238. · Zbl 1391.53050 |
| [9] | R. Espinola, A. Nicolae, Mutually nearest and farthest points of sets and the drop theorem in geodesic spaces, Monatsh. Math., 165(2012), 173-197. · Zbl 1238.41017 |
| [10] | R. Espinola, B. Piatek, Fixed point property and unbounded sets in CAT(0) spaces, J. Math. Anal. Appl., 408(2013), 638-654. · Zbl 1306.47063 |
| [11] | A. Fernandez Leon, A. Nicolae, Best proximity pair results relatively nonexpansive mappings in geodesic spaces, Numer. Funct. Anal. Optim., 35(2014), 1399-1418. · Zbl 1301.54066 |
| [12] | M. Gabeleh, Common best proximity pairs in strictly convex Banach spaces, Georgian Math. J., 24(2017), 363-372. · Zbl 1453.47007 |
| [13] | M. Gabeleh, Convergence of Picard’s iteration using projection algorithm for noncyclic con-tractions, Indag. Math., 30(2019), 227-239. · Zbl 1470.47064 |
| [14] | M. Gabeleh, Corrigendum to the paper “Equivalence of the Existence of Best Proximity Points and Best Proximity Pairs for Cyclic and Noncyclic Nonexpansive Mappings”, Demon. Math., 54(2021), 9-10. · Zbl 1470.47037 |
| [15] | M. Gabeleh, H.P. Künzi, Condensing operators of integral type in Busemann reflexive convex spaces, Bull. Malays. Math. Sci. Soc., doi: (2020), 10.1007/s40840-019-00785-x. · Zbl 1434.53043 · doi:10.1007/s40840-019-00785-x |
| [16] | M. Gabeleh, S.P. Moshokoa, Study of minimal invariant pairs for relatively nonexpansive map-pings with respect to orbits, Optimization, (2020), doi: 10.1080/02331934.2020.1745797. · Zbl 1475.90127 · doi:10.1080/02331934.2020.1745797 |
| [17] | M. Gabeleh, O. Olela Otafudu, Nonconvex proximal normal structure in convex metric spaces, Banach J. Math. Anal., 10(2016), 400-414. · Zbl 1335.54042 |
| [18] | M. Gabeleh, O. Olela Otafudu, Markov-Kakutani’s theorem for best proximity pairs in Hadamard spaces, Indag. Math., 28(2017), 680-693. · Zbl 1367.54027 |
| [19] | K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984. · Zbl 0537.46001 |
| [20] | S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150. · Zbl 0286.47036 |
| [21] | V. Sankar Raj, A best proximity point theorem for weakly contractive non-self-mappings, Non-linear Anal., 74(2011), 4804-4808. · Zbl 1228.54046 |
| [22] | H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16(1991), 1127-1138. · Zbl 0757.46033 |
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